Topographic Disturbance in Viscous Shear Flow

Abstract

The initial-value problem corresponding to perturbed viscous shear flow in shallow water over topography is solved both analytically and numerically. A formal solution is obtained analytically by using the Fourier-Laplace transform. On the other hand, a numerical solution is obtained for Froude number Fr=0.1 and a basic flow U=tanh(y) by time integration. Both spatial and temporal behavior of the solution are studied.
The stability of shear flows which are unstable in an inviscid fluid over a flat bottom changes with the strength of the friction; it varies from unstable to stable through a resonance between the topographic forcing and a barotropic wave. The structure of the disturbance is very similar to the unstable barotropic wave as long as the friction is slightly greater than the resonance point. It is supplied with energy from the basic shear flow through the Reynolds stress. Furthermore, a vortex remains in the basic shear zone when the topography moves across the flow. The structure of this vortex is also similar to the unstable barotropic wave and its energy is supplied from the shear flow. Thus, the vortex has a long life time against the friction.
If the friction is large, the disturbance directly reflects the topographic forcing. The structure is similar to gravity waves and the energy is supplied from the topographic forcing.
A comparison with the vortex observed in the atmosphere is also described.